Tuning order in the cuprate superconductors Eugene Demler

Tuning order in the cuprate superconductors Eugene Demler (Harvard) Kwon Park Anatoli Polkovnikov Subir Sachdev Matthias Vojta (Augsburg) Ying Zhang Science 286, 2479 (1999). Transparencies online at http: //pantheon. yale. edu/~subir

Parent compound of the high temperature superconductors: Mott insulator: square lattice antiferromagnet Ground state has long-range magnetic (Néel) order Néel order parameter:

Introduce mobile carriers of density d by substitutional doping of out-of-plane ions e. g. Exhibits superconductivity below a high critical temperature Tc Quantum numbers of ground state and low energy quasiparticles are the same, but characteristics of the Mott insulator are revealed in the vortices. S. Sachdev, Phys. Rev. B 45, 389 (1992); K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001). STM measurement of J. E. Hoffman et al. , Science, Jan 2002.

Zero temperature phases of the cuprate superconductors as a function of hole density Neel LRO SDW along (1, 1) +localized holes SC+SDW SC ~0. 12 ~0. 05 H Theory for a system with strong interactions: describe SC and SC+SDW phases by expanding in the deviation from the quantum critical point between them. B. Keimer et al. Phys. Rev. B 46, 14034 (1992). S. Wakimoto, G. Shirane et al. , Phys. Rev. B 60, R 769 (1999). G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook, J. Kulda, Science 278, 1432 (1997). Y. S. Lee, R. J. Birgeneau, M. A. Kastner et al. , Phys. Rev. B 60, 3643 (1999). J. E. Sonier et al. , cond-mat/0108479. C. Panagopoulos, B. D. Rainford, J. L. Tallon, T. Xiang, J. R. Cooper, and C. A. Scott, preprint. d

Insulator with localized holes Further neighbor magnetic couplings T=0 Universal properties of magnetic quantum phase transition change little along this line. Magnetic order Experiments SC+SDW Superconductor (SC) Concentration of mobile carriers d in e. g. S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, 11919 (1994)

Outline I. Magnetic ordering transitions in the insulator (d=0). II. Theory of SC+SDW to SC quantum transition III. Phase diagrams of above in an applied magnetic field Comparison of predictions with experiments. IV. Conclusions

I. Magnetic ordering transitions in the insulator Square lattice with first(J 1) and second (J 2) neighbor exchange interactions (say) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). O. P. Sushkov, J. Oitmaa, and Z. Weihong, Phys. Rev. B 63, 104420 (2001). M. S. L. du Croo de Jongh, J. M. J. van Leeuwen, W. van Saarloos, Phys. Rev. B 62, 14844 (2000). Neel state Spin-Peierls (or plaquette) state “Bond-centered charge order” See however L. Capriotti, F. Becca, A. Parola, S. Sorella, condmat/0107204.

Properties of paramagnet with bond-charge-order Stable S=1 spin exciton – quanta of 3 -component Spin gap S=1/2 spinons are confined by a linear potential. Develop quantum theory of SC+SDW to SC transition driven by condensation of a S=1 boson (spin exciton)

Further neighbor magnetic couplings Framework for spin/charge order in cuprate superconductors Confined, paramagnetic Mott insulator has 1. Stable S=1 spin exciton . 2. Broken translational symmetry: - bondcentered charge order. 3. S=1/2 moments near non-magnetic impurities Experiments Magnetic order T=0 Concentration of mobile carriers d Theory of magnetic ordering quantum transitions in antiferromagnets and superconductors leads to quantitative theories for • Spin correlations in a magnetic field • Effect of Zn/Li impurities on collective spin excitations

II. Theory of SC+SDW to SC quantum transition Spin density wave order parameter for general ordering wavevector Wavevector K=(3 p/4, p) Exciton wavefunction Fa(r) describes envelope of this order. Phase of Fa(r) represents sliding degree of freedom Associated “charge” density wave order J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989). H. Schulz, J. de Physique 50, 2833 (1989). O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. B 57, 1422 (1998).

Action for SDW ordering transition in the superconductor Similar terms present in action for SDW ordering in the insulator Coupling to the S=1/2 Bogoliubov quasiparticles of the d-wave superconductor Trilinear “Yukawa” coupling is prohibited unless ordering wavevector is fine-tuned.

Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field B. Lake, G. Aeppli, K. N. Clausen, D. F. Mc. Morrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. Schröder, Science 291, 1759 (2001).

S. Sachdev, Phys. Rev. B 45, 389 (1992), and N. Nagaosa and P. A. Lee, Phys. Rev. B 45, 966 (1992), suggested an enhancement of dynamic spin-gap correlations (as in a spin-gap Mott insulator) in the cores of vortices in the underdoped cuprates. In the simplest mean-field theory, this enhancement appears most easily for vortices with flux hc/e. D. P. Arovas, A. J. Berlinsky, C. Kallin, and S. -C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) suggested static Néel order in the cores of vortices (SC order “rotates” into Néel order in SO(5) picture). Using a picture of “dynamically fluctuating spins in the vortices”, the amplitude of the field-induced signal, and the volume-fraction of vortex cores (~10%), Lake et al. estimated that in such a model each spin in the vortex core would have a lowfrequency moment equal to that in the insulating state at d=0 (0. 6 m. B). Observed field-induced signal is much larger than anticipated.

III. Phase diagrams in a magnetic field. Insulator with localized holes T=0 A. Effect of magnetic field on onset of SDW on insulator B. Effect of magnetic field on Further neighbor magnetic couplings SC+SDW to SC transition Magnetic order SC+SDW Superconductor (SC) Concentration of mobile carriers d in e. g.

III. Phase diagrams in a magnetic field. A. Effect of magnetic field on onset of SDW in the insulator H SDW Spin singlet state with a spin gap J 2/J 1 Characteristic field gm. BH = D, the spin gap 1 Tesla = 0. 116 me. V Related theory applies to spin gap systems in a field and to double layer quantum Hall systems at n=2

III. Phase diagrams in a magnetic field. (extreme Type II superconductivity) B. Effect of magnetic field on SDW+SC to SC transition Infinite diamagnetic susceptibility of non-critical superconductivity leads to a strong effect. • Theory should account for dynamic quantum spin fluctuations • All effects are ~ H 2 except those associated with H induced superflow. • Can treat SC order in a static Ginzburg-Landau theory

Envelope of spin-exciton eigenmode in potential V 0(x) Energy Spin gap D 0 x Vortex cores

Dominant effect: uniform softening of spin uniform excitations by superflow kinetic energy

Main results T=0 • All functional forms are exact. E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

Structure of long-range SDW order in SC+SDW phase Computation in a self-consistent “large N” theory Y. Zhang, E. Demler, and S. Sachdev, cond-mat/0112343 s = sc s – sc = -0. 3

Neutron scattering measurements of static spin correlations of the superconductor+spin-density-wave (SC+SDW) in a magnetic field

B. Lake, G. Aeppli, et al. , Nature, Jan 2002.

Prediction of static CDW order by vortex cores in SC phase, with dynamic SDW correlations “Spin gap” state in vortex core appears by a “local quantum disordering transition” of magnetic order: by our generalized phase diagram, charge order should appear in this region. K. Park and S. Sachdev Physical Review B 64, 184510 (2001).

Pinning of static CDW order by vortex cores in SC phase, with dynamic SDW correlations A. Polkovnikov, S. Sachdev, M. Vojta, and E. Demler, cond-mat/0110329 Y. Zhang, E. Demler, and S. Sachdev, cond-mat/0112343 Superflow reduces energy of dynamic spin exciton, but action so far does not lead to static CDW order because all terms are invariant under the “sliding” symmetry: Small vortex cores break this sliding symmetry on the lattice scale, and lead to a pinning term, which picks particular phase of the local CDW order With this term, SC phase has static CDW but dynamic SDW ; “Friedel oscillations of a doped spin-gap antiferromagnet”

Pinning of CDW order by vortex cores in SC phase Computation in self-consistent large N theory

Simplified theoretical computation of modulation in local density of states at low energy due to CDW order induced by superflow and pinned by vortex core A. Polkovnikov, S. Sachdev, M. Vojta, and E. Demler, cond-mat/0110329

(E) STM around vortices induced by a magnetic field in the superconducting state J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science, Jan 2002 Local density of states 1Å spatial resolution image of integrated LDOS of Bi 2 Sr 2 Ca. Cu 2 O 8+d ( 1 me. V to 12 me. V) at B=5 Tesla. S. H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).

Vortex-induced LDOS integrated from 1 me. V to 12 me. V 7 p. A b 0 p. A 100Å J. Hoffman et al, Science, Jan 2002.

Fourier Transform of Vortex-Induced LDOS map K-space locations of vortex induced LDOS K-space locations of Bi and Cu atoms Distances in k –space have units of 2 p/a 0 a 0=3. 83 Å is Cu-Cu distance J. Hoffman et al Science, Jan 2002.

Why does the charge order have period 4 ? “Large N” theory in region with preserved spin rotation symmetry S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999). M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000). See also J. Zaanen, Physica C 217, 317 (1999), S. Kivelson, E. Fradkin and V. Emery, Nature 393, 550 (1998), S. White and D. Scalapino, Phys. Rev. Lett. 80, 1272 (1998).

Effect of magnetic field on SDW+SC to SC transition (extreme Type II superconductivity) T=0 Main results Neutron scattering observation of SDW order enhanced by superflow. Neutron scattering observation of SDW fluctuations enhanced by superflow. STM observation of CDW fluctuations enhanced by superflow and pinned by vortex cores. Prospects for studying quantum critical point between SC and SC+SDW phases by tuning H ?

Consequences of a finite London penetration depth (finite k)

(A) Spatially resolved NMR of Zn/Li impurities in the superconducting state Li Inverse local susceptibilty in YBCO 7 Li NMR below T J. Bobroff, H. Alloul, W. A. Mac. Farlane, P. Mendels, N. Blanchard, G. Collin, and J. -F. Marucco, Phys. Rev. Lett. 86, 4116 (2001). A. M Finkelstein, V. E. Kataev, E. F. Kukovitskii, G. B. Teitel’baum, Physica C 168, 370 (1990). c

Neutron scattering measurements of phonon spectra k = 0 Discontinuity in the dispersion of a LO phonon of the O ions at wavevector k = p/2 : evidence for bond-charge order with period 2 a k = p La 1. 85 Sr 0. 15 Cu. O 4 YBa 2 Cu 3 Cu. O 6. 95 R. J. Mc. Queeney, T. Egami, J. -H. Chung, Y. Petrov, M. Yethiraj, M. Arai, Y. Inamura, Y. Endoh, C. Frost and F. Dogan, condmat/0105593. R. J. Mc. Queeney, Y. Petrov, T. Egami, M. Yethiraj, G. Shirane, and Y. Endoh, Phys. Rev. Lett. 82, 628 (1999). L. Pintschovius and M. Braden, Phys. Rev. B 60, R 15039 (1999).

G. Aeppli, T. E. Mason, S, M. Hayden, H. A. Mook, and J. Kulda, Science 278, 1432 (1998).

Neutron scattering in YBCO S=1 exciton near antiferromagnetic ordering wavevector Q = (p, p) Resolution limited width H. F. Fong, B. Keimer, D. Reznik, D. L. Milius, and I. A. Aksay, Phys. Rev. B 54, 6708 (1996)

Quantum dimer model – D. Rokhsar and S. Kivelson Phys. Rev. Lett. 61, 2376 (1988) Quantum “entropic” effects prefer one-dimensional striped structures in which the largest number of singlet pairs can resonate. The state on the upper left has more flippable pairs of singlets than the on the lower left. These effects lead to a broken square lattice symmetry near the transition to the Neel state. N. Read and S. Sachdev Phys. Rev. B 42, 4568 (1990).

Spatially resolved NMR around vortices induced by a magnetic field in the superconducting state Nature, 413, 501 (2001).

Full solution of self-consistent large N equations for phases and phase boundaries Y. Zhang, E. Demler, and S. Sachdev, cond-mat/0112343

Full solution of self-consistent large N equations for phases and phase boundaries Y. Zhang, E. Demler, and S. Sachdev, cond-mat/0112343.

Dynamic SDW fluctuations in the SC phase Field H chosen to place the system close to boundary to SC+SDW phase

Doping the paramagnetic Mott insulator “Large N” theory in region with preserved spin rotation symmetry S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999). M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000). See also J. Zaanen, Physica C 217, 317 (1999), S. Kivelson, E. Fradkin and V. Emery, Nature 393, 550 (1998), S. White and D. Scalapino, Phys. Rev. Lett. 80, 1272 (1998).